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Θεοδόσιος Αλεξιάδης
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8 l 0 l 2 l 1 l 1 l 1

10 l 2 l 2 l 1 l p λ λ µ

19 R N l 2 R N l 2 2 = N x i l p p R N l p N p = ( x i p ) 1 p i=1 l 2 l p p = 2 l p l 1 R N l 1 i=1 x 2 i 1 = N x i i=1 l p p p R N l 0 0 = {i x i 0} R N s s 0 s

20 R N Φ N N R N R N s Φ = Φ s Φ Ψ M R N Ψ M N M < N = Ψ = Ψ = (ΨΦ) = A A Ψ ΨΦ R M = A +

21 A A δ s s δ s s (1 δ s ) 2 2 A 2 2 (1 + δ s) 2 2 δ s s l 0 l 0 l 0 ˆ = 0 : = A R N l A( 1 2 ) = A 1 A 2 = = 0

22 l 0 s l 0 l = 2s = 0 l 0 2s 0 : A = 0 0 > 2s 2s A A (A) = 0 : A = 0 A A ex = (A ex ) = 3 (A ex ) = 2 (A ex ) R N s A (A) > 2s = A 0 : = A x

23 N, M A R M M M + 1 (A) M + 1

24 A f E ( ) = 1 ( 2πσe) 2 M ( 2 2 ) 2σ 2 e A

25 A f Y ( ) = 1 ( 2πσe) ( A 2 2 ) 2 M 2σ 2 e f Y ( ) ˆ = A 2 2 R N l 2 A T Aˆ = A T A T A ˆ = (A T A) 1 A T M < N = A

26 Fitting a quadratic model a =2,b = 3,c =0 Measuruments Real underlying model Fitted 1 st order model Fitted 2 nd order model Fitted 15 th order model 300 y x y = ax 2 + bx + c

27 P ( ) = P (, ) P ( ) = P ( )P ( ) P ( ) P ( ) ˆ = P ( )P ( ) R N P ( ) P ( ) P ( ) l 2 l 2 P ( ) = δ( A )

28 ˆ = R N 1 δ( A ) ( 2πσ x ) M ( 2σx 2 ) = A l 2 ˆ = R N 2 : = A = A M < N 2 2 ˆ = A T (AA T ) 1 ˆ = R N 2 A 2 2σe σx 2 ˆ = A λ 2 2 R N λ = σ2 e σ 2 x

29 Impulse Response Estimation Least square norm fitting of sparse system 20 Meas. 40 Meas. 60 Meas n(time)

30 λ λ λ λ λ ˆ = (A T A + λi N ) 1 A T I n N N l 2

31 l 2 l 2 l 2 a, b 1 = [a + b, 0] T 2 = [a, b] T = 2ab l 2 l l 2 X x λx P (X > x) = λ X 1,... X n N P ( X i > x) P ( X i > x) x i=1...n i=1

32 x X Y = (X) f(x) = 1 xσ 2π (( x µ)2 2σ 2 ) ν 1 t2 ν+1 f T (t) = νb( 1 2, ν + ) 2 2 )(1 ν B(x, y) l 1 l 2 l 1 x R f(x; µ, b) = 1 µ ( x ) 2b b l 1 R N b µ f( ; µ, b) = ( 1 2b )N ( µ 1 ) b

33 Probability density Comparisson Student with Normal Student n =2 Studnet n =5 Student n =10 Normal x n ˆ = 1 : = A R N ˆ = A λ 1 R N λ = 2σ2 e b

34 Comparisson Laplace with Normal Laplace b = Normal Probability density x b = 2 2 b

35 l 1 l 2 l 1 R 2 = A A y = A 11 x 1 + A 12 x 2 y x 1 x 2 2 = c 1 = c c c 2c c c l 2 c l 1 A A l p p 1 p p

36 2.0 l 2 norm minimization 2.0 l 1 norm minimization x 2 x x x 1 x 2 = 5 5 x 2 = 1 x 1 2 x 1, x 2 y = x 1 + 2x 2 = 1 1 = 1 2

37 l 2 l p p > 1 p p p = l p norm comparison 0.5 x p =0.5 p =1 p = x = 1 1 = 1 2 = 1 p = 0.5 l 0.5

38 l 1 l 2 l 0 l 1 l 0 l 2 l 0 A M N i i µ(a) A µ(a) = 1 i<j N T i j i 2 j 2 A (A) (A) µ(a) l 0

39 R N 0 < 1 2 (1 + 1 µ(a) ) = A = A 0 < 1 2 (1 + 1 µ(a) ) 1 2 A l 0 l 0 l A 1 A s 2s (1 δ 2s ) A( 1 2 ) 2 2 (1 + δ 2s) δ 2s s A

40 l 1 δ 2s < 2 1 s l 0 l 0 δ 2s < 1 A N (0, 1 M ) R M s δ s M Cs (N/s) C δ s A

41 l 2 u i N u i : i=1 = A N (0, 1) N (0, 1 M ) = A ( ) ( ) = N 0 N

42 x 1.0 Perfect reconstruction with basis pursuit n y 3 Samples generated by random gaussian matrix n

43 SNR vs sparsity with varying number of measurements SNR Noiseless SNR SNR 20db sparsity sparsity A M

44 A l 2 ˆ = 1 : A 2 2 e2 R N

45 λ e λ e Aˆ e l 1 ˆ = A 2 2 : 1 l R N l l 1 e l 2 λ s s δ 2s < C 0 s 1 2 s 1 + C 1 e s

46 SNR vs sparsity with varying number of measurements SNR SNR 20db SNR SNR 10db sparsity sparsity A M

47 l 1 l 0 l 1 l 2 l 0 A

48 = A A T = A T A A T 0 = 0 = i i k k i j i = T k i 1 1 k N i = i 1 ( T j i i 1 ) ji j i i = i 1 + T j i i 1 δ N (j i ) δ N (j i ) R N j i A i 1 = ( T ) +

49 T = 0 A S 0 = i S i = S i {j i } A Si A S i ˆ i = A Si 2 R i i i = A i

50 l 1 = A 0 < 1 2 (1 + 1 µ(a) )

51 SNR vs sparsity with varying number of measurements SNR Noiseless SNR SNR 20db sparsity sparsity A M

52 t t A T i 1 s s i S i i 1 t A T i 1 A Si ˆ i i = H s (ˆ i ) H s s i = A i

53 H s = [1, 8, 9, 0] T H 2 ( ) = [0, 8, 9, 0] T t t = s t = 2s s t = s s t = s

54 SNR vs sparsity with varying number of measurements SNR Noiseless SNR SNR 20db sparsity sparsity A M

55 g(x) = 0 f(x) = x f x i+1 = f(x i ) f = A i i i i = i + µa T i µ i = A i = A 1 = µa T = µa T (A )

56 1 µ µ T i+1 = T ( i + µa T i ) i+1 = T i ( i + µa T i ) µ i+1 = T i ( i + µ i A T i ) l 0 H s µ A 2 < 1 A T A µ

58 SNR vs sparsity with varying number of measurements SNR Noiseless SNR SNR 20db sparsity sparsity M

59 α α α α S α α Soft vs Hard Thresholding 0.6 Soft Thresholding 1.0 Hard Thresholding S H input input i+1 = S λµ ( i + µa T i )

60 µ A T A i = S λµ ( i + µa T ( A i )) 0 = 0 i+1 = i + t i 1 t i+1 ( i i 1 ) t i t i+1 = t 2 i 2 t 1 = 1

63 (n) = [h 0 (n),..., h p 1 (n)] p x(n) y(n) d(n) = y(n) + v(n) v(n) (n) ˆ x(n) d(n) = p (n) = [x(n),..., x(n p + 1)] T y(n) = T (n) (n) (n) ˆ (n) ˆ (n)

64 ˆ (n 1) ŷ(n) ŷ(n) = ˆ T (n 1) (n) e(n) = d(n) ŷ(n) = d(n) ˆ T (n 1) (n) J(n) = n λ n i e 2 (i) i=0 λ (0, 1] λ λ λ = 1 R xx (n)ˆ (n) = dx (n) R xx (n) λ R xx (n) = n λ n i (i) T (i) i=0

65 (n) x(n) d(n) dx (n) = n λ n i (i)d(i) i=0 R xx (n) dx R xx (n) = λr xx (n 1) + (n) T (n) dx (n) = λ dx (n 1) + (n)d(n) O(p 3 ) p P (n) = R xx (n) 1 e(n) = d(n) T (n 1) (n) (n) = P (n 1) (n) λ + T (n)p (n 1) (n) P (n) = λ 1 (P (n 1) (n) T (n)p (n 1)) ˆ (n) = ˆ (n 1) + e(n) (n) O(p 2 ) ˆ = P (0) = δ 1 I p δ I p p p

66 λ λ v(n) λ λ λ λ λ J(n) = E{e 2 (n)} J(n) µ ˆ (n) = ˆ (n 1) µ 2 ˆ J(n)

67 Error to Signal Ratio Impulse Response estimation under noise λ =0.10 λ =0.50 λ = n(iterations) λ [1, 2, 3, 4] λ λ λ

68 Impulse Response estimation of changing system λ =0.10 λ =0.50 λ =1.00 Error to Signal Ratio n(iterations) λ [1, 2, 3, 4] T [ 3, 4, 7, 8] T λ = 1

69 ˆ (n) = ˆ (n 1) + µe{ (n)e(n)} E{ (n)e(n)} ν Ê{ (n)e(n)} = 1 ν 1 (n i)e(n i) ν i=0 ν ν ν = 1 ˆ (n) = ˆ (n 1) + µ (n)e(n) O(p) µ λ i R xx 2 0 < µ < i λ i ˆ (n) = ˆ (n 1) + µ (n)e(n) (n) T (n)

70 Error to Signal Ratio Impulse Response estimation µ =0.10 µ =1.00 µ =1.50 µ =1.99 µ = n(iterations) [1, 2, 3, 4] T µ = 1 µ = 2 µ = 1 µ

71 ν i Ji n(ˆ ) ˆ i N i ψ i (n) = ˆ i (n 1) µ i c l,i ˆ J l n (ˆ l (n 1)) l N i µ i c l,i l i l i N i c l,i = 1 l N i µ i i ˆ i (n) = l Ni a l,i ψ i (n) a l,i l i c l,i

72 a l,i = 1 l N i c i,i = 1 a i,i = 1

73 y(n) θ p( ) = p( θ)p(θ )dθ p(θ )

74 α α α A s (n) R s (n) R(n) A s (n 1) R s (n) = R(n) + A s (n 1) A s α λ

75 b (n) s s H s (n) = ˆ (n) + e(n) (n) T (n) (n) H s (n) ˆ (n + 1) = H s ( (n))

76 5 0-5 RLS GARLS ASVB-S ASVB-L ASVB-mpL NMSE(dB) Iterations (n) λ

77 SNR(db) Sparsity aware NLMS NLMS Iterations

78 ˆ (n) n s q = { h i h i 0} ĥ(n) h 2 2 q2 2 n 0 N (H s (ˆ (n))) = ( ) n n 0 n 0 N (H s (ˆ (n)) = ( ) n n 0 ( ) (H s (ˆ (n)) z ( ) h z q q (H s (ˆ (n)) ϕ(n) > ĥz(n) ϕ(n) s ˆ (n) (H s (ˆ (n)) ( ) r ( ) h r = 0 (H s (ˆ (n)) ĥr(n) ϕ(n) ĥr(n) > ĥz(n) ϵ ĥ r (n) 2 = ĥz(n) 2 + ϵ 2 g(n) = ˆ (n) 2 2 g(n) ĥr(n) h r 2 + ĥz(n) h z 2 h r = 0 ĥ r (n) 2 = ĥz(n) 2 + ϵ 2 g(n) 2ĥz(n) 2 2ĥz(n)h z + h 2 z + ϵ 2 f(x) = 2x 2 2h z x+(h 2 z+ϵ 2 ) f = 0 x = h z/2 x 2 h z 2 h z 2 + h 2 z + ϵ 2 = ϵ 2 + hz2 2 n g(n) ϵ 2 + h z 2 2 > h 2 z 2 q2 2

79 g(n) = ĥ(n) h 2 2 q2 2 ˆ (n) ϕ(n) (H s (ˆ (n)) s (H s (ˆ (n))) ( ) r, z r z (H s (ˆ (n))) ( ) ( ) (H s (ˆ (n))) ξ(n) ϕ(n) s k (n) k > s ˆ (n) n s q = { h i h i 0} ĥ(n) h 2 2 q 2 (1 1 τ+2 ) τ = k s n 0 N (H k (ˆ (n))) ( ) n n 0

80 SNR(db) Sparsity aware NLMS NLMS Iterations

81 n 0 N (H k (ˆ (n)) ( ) n n 0 ( ) (H k (ˆ (n)) z ( ) h z q q (H k (ˆ (n)) ξ(n) > ĥz(n) τ + 1 (H k (ˆ (n)) ( ) r i ( ) h ri = 0 (H k (ˆ (n)) ĥr i (n) ξ(n) ĥr i (n) ĥz(n) ϵ i ĥ ri (n) 2 = ĥz(n) 2 + ϵ 2 i g(n) = ˆ (n) 2 2 τ + 2 τ+1 g(n) ĥr i (n) h ri 2 + ĥz(n) h z 2 i=1 h ri = 0 ĥ ri (n) 2 = ĥz(n) 2 + ϵ 2 i ϵ2 t = τ+1 i=1 ϵ2 i g(n) (τ + 2)ĥz(n) 2 2ĥz(n)h z + h z 2 + ϵ 2 t f(x) = (τ +2)x 2 2h z x+(h 2 z +ϵ 2 t ) f x = 0 x = h z/(τ + 2) h z 2 τ+2 2 h z 2 τ+2 + h z 2 + ϵ 2 t = ϵ 2 t + h 2 z (1 1 τ+2 ) n g(n) ϵ 2 t + h z 2 (1 1 τ + 2 ) > h z 2 (1 1 τ + 2 ) q2 (1 1 τ + 2 ) g(n) = ĥ(n) h 2 2 q 2 (1 1 τ+2 )

82 25 20 SNR(db) Sparsity aware NLMS NLMS Iterations

83 ν a i,j i j a i,i a i,j = a ν 1 a i,i = 1 a i j a x j (n) j n j (n) = [x j (n),..., x j (n p+1)] T j d j (n) e j (n) j (n) = ˆ j (n) + e j(n) j (n) T j (n) j(n) q j j (n) = (1 a) j (n) + a q(n) ν 1 ˆ j (n + 1) = H s ( j (n)) a

84 α = 0.4 α = 0.4

85 SNR(db) Sparsity aware NLMS NLMS Iterations α

87 x(t) t x j (t) j t x j (t) = x(t t dj ) t > t dj t dj j x j (t) = x(t t dj ) + w j (t) t > t dj i j x i (t) = x j (t t di,j ) + w i,j (t) t di,j w i,j t di,j = t di t dj

88 d i i d i d j = t di,j c c cos(θ i,j ) θ i,j cos(θ i,j ) = t d i,j c d i,j di, j t = t di,j i j

89 i j i,j N h j = X i i,j j h j X i i Q Q j = QX i i,j Q QX i X i ˆ i,j = Q j QX i λ 1 R N h ˆ i,j

90 Delay estimation via sparsity 0.5 Whole estimation 0.5 Zoomed in ĥ ĥ Time delay(samples) Time delay(samples) N h = 4097

91 1.6 Zoomed in ĥ Time delay(samples) ω 0 F ω ( i )(ω 0 ) = e jω 0t di F ω ( )(ω 0 ) F ω θ

92 ˆ t di = t d0 + t di,0 = t d0 + d i,0cos(θ) c F ω ( i )(ω 0 ) = α i (θ)f ω ( )(ω 0 ) α i (θ) θ N s sk θ k N s F ω ( i )(ω 0 ) = α i (θ k )F ω ( sk )(ω 0 ) k=1 F ω ( sk )(ω 0 ) (ω0 ) α i (θ k ) α i F ω ( i )(ω 0 ) = α T i (ω0 ) F ω ( i )(ω 0 ) (ω0 ) α i A (ω0 ) (ω0 ) = A (ω0 ) (ω0 )

93 N ω (ωl ) (ωl ) A (ωl ) A A = A (ω0 ) A (ω1 ) = A... A (ωnω ) N θ θ k N ω ω l N θ N ω N ω θ k (k) (k) (k) 2 l 1 ˆ = C NωN θ N θ A λ (k) 2 2 k=1

94 0 Angle of arrival detection via sparsity Normalized Power(db) θ(angle)

95 = Φ Φ

96 U y = U = (UΦ) σ σ = 1

97 500 Original Spectrum 500 Estimated Spectrum Amplitude Amplitude Frequency in Hz Frequency in Hz 10 6

98 200 Original Spectrum 200 Estimated Spectrum Amplitude Amplitude Frequency in Hz Frequency in Hz 10 6

99 200 Original Spectrum 140 Estimated Spectrum Amplitude Amplitude Frequency in Hz Frequency in Hz 10 6

100

101 l 0 l 2 l 1 l 1 l 0 l 1

102

103

104

105

106

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